How to step-select vertices to side?
I don't think a simple way exists yet. 

A workaround could be to use vertex groups. By selecting a centre vertex and then selecting side of active to select all the vertices on the side you don't want, you can then create a vertex group and assign these vertices to the group.

With the vertex group setup you can select the centre vertices and expand your selection, then deselect the vertices in the group, which will leave you with only vertices on one side of your mesh selected.



I think you may even want to select only one new edge loop each time, in that case, after deselecting the vertices to one side you can assign the newly selected vertices to the group, then the next time you expand the selection and deselect you will also deselect the previous loop and all the previously selected vertices.



As you are saving your progress each time you can perform any steps you want between each step, you can always select the group vertices and expand then deselect to get the next loop.